I'm taking a geometry course and had a question. There's a section where we prove that lines are the shortest path connecting two points on a curve, and the proof for that assumes that the curve is continuous and differentiable.
I'm wondering, though, why do we have to assume this? Would it be impossible to make the proof if the curve were not differentiable?
If there's any resources or material out there that I've missed or could use to study, it'd be of great help if anyone could point it out for me. Thanks in advance!
First, if you allowed discontinuous "paths", you'd get useless geodesics. For example, there would be a path of length 2 meters from where I'm sitting to wherever you are. The first meter would be from my chair to the nearby bookshelf, and then there would be a discontinuous jump to a point one meter from your location, from where the second meter of the path can reach you.
OK, so discontinuity is a bad idea here. But why does one want differentiability? As far as I can see, the point is that to talk about a "shortest path" we need a definition of the length of a path. Differentiability provides that, with the usual integral formula for arc length.
It is possible to define length for some (but not all) non-differentiable curves. The definition involves approximating the curve with a polygonal curve whose edges are chords of the given curve, and then taking the limit as the number of edges increases while the lengths of individual edges approach $0$. Curves for which this process gives a finite length in the limit are called rectifiable curves. I wouldn't be at all surprised if there's a good theory of geodesics that deals with continuous rectifiable curves without requiring differentiability. (There might even be a theorem saying that, if the space you're working in is nice enough, then geodesics defined under these weaker hypotheses will automatically be differentiable.) Unfortunately, I don't know that theory, so I'll stop writing and wait for an expert to come along.